In the comments on Ralph’s post, Julien raised the issue of Glimpse cases as a problem for adherence conditions on knowledge, in response to my question about whether a probabilistic adherence condition would be advantageous to use. Glimpse cases, such as glimpsing a painted bunting in my live oak tree as I happen to look up from writing this post, present challenges to adherence conditions, since one is highly likely not to have noticed the bird at all.

I think Roush’s probabilistic truth-tracking theory is much superior to the original Nozick theory, so wanted to post on whether her account can handle Glimpse cases. The feature of her theory that must do the work here is the condition on what gets held fixed when assessing the relevance condition and the adherence condition, which are, respectively:

P(~B(p)|~p) is very high.

P(B(p)|p) is very high and P(B(~p)|p) is very low.

I’ll put the details of what gets held fixed and how this helps with Glimpse cases below the fold, but the interesting point here is that the fixing condition allows the conclusion that Roush’s adherence condition is satisfied in Glimpse cases.

So, first the technical details of what gets held fixed. What gets held fixed is the actual (objective) probability of a claim, given (and only given) that it meets either of two inequality conditions. I’ll given these formally, and then give Roush’s informal gloss on them. The formal inequalities are these:

For all q such that q is not a conjunction or a universal generalization nor B(p), ~B(p), B(p), ~B(~p); hold Pr(q) fixed at its actual value iff:

* $$|Pr(q|\sim\!p) – Pr(q|p)| \leq |Pr(\sim\!p|q) – Pr(p|q)|$$ and $$|Pr(\sim\!q|\sim\!p) – Pr(\sim\!q|p)| \leq |Pr(\sim\!p|q) – Pr(p|q)|$$

or

** $$|Pr(q|p) – Pr(q|\sim\!p)| < |Pr(p|q) - Pr(p|\sim\!q)|$$ and $$|Pr(\sim\!q|p) - Pr(\sim\!q|\sim\!p)| < |Pr(p|q) - Pr(p|\sim\!q)|$$. Roush glosses these conditions as follows: "Criterion * says that p's being false rather than true makes less or the same difference to q's being true, and to q's being false, than the difference that q's being true or false makes to p's being false." For criterion **, she says, "[H]ere we want to allow variation in those matters that the truth value of p makes no difference to, and allow variation in those things that the truth of p makes more of a difference to than they make to p." The condition useful in responding to Glimpse cases is criterion *, and notice the presence of the painted bunting in my tree makes no probabilistic difference to whether I glance in the way I did or whether I didn't glance at all. Notice as well that whether I glance in the way I did, or whether I didn't glance, makes no difference to whether there is a painted bunting in my live oak tree. So, criterion * is satisfied in this case for the claim about the particularities of my glimpse (q), because p's being false rather than true makes the same difference to the truth value of q as the truth value of q makes to p. So the actual probability of q gets held fixed when assessing the adherence condition, and the actual probability of the particularities of my glimpse is, by hypothesis, 1. So, the probability involved in the adherence condition is the probability of my believing that there is a painted bunting in my live oak tree, given the presence of the bird and the particularities of my glimpse. This probability strikes me as plausibly held to be very high. (I will here assume minimal logical coherence for my own belief states, so the probability of believing not-p given p will be correspondingly low--this assumption is an important one in Roush's theory, and generates independent concerns about the theory, but since they are independent of the worries caused by Glimpse cases for adherence conditions on knowledge, I'll ignore them here. If you wish further info on that issue, see Horacio's review of Sherri's book at NDPR). So, bottom line: it looks like Roush's fixing condition gives a decent answer to how the adherence condition is satisfied in Glimpse cases.

Jon: thanks a lot for expanding this point. I need to think about it again.

A couple of quick questions. A slight modification of the case will help. Suppose there’s a branch opposite my windows; birds regularly pass by, stop a second on the branch and fly away. Often, but not always, they tweet when on the branch. Sometimes, but not always, the tweeting attracts my attention and makes me glance their way.

My first question is the following: is “getting fixed” not closed under entailment? It looks like “a bird stopping on the branch” affects the probability that I glance their way; but “a bird silently stopping on the branch” doesn’t. So it looks like the second is kept fixed, it entails the first, but the first does not get fixed. But it doesn’t make sense to evaluate how the probabilities would be if it was certain that a bird silently stopped on the branch but yet open that no bird stopped on the branch.

The second is whether Roush’s adherence can handle that modified Glimpse case. It looks like (a) the probability that I glance is affected by whether there’s a bird outside, (b) thus the fact that I glance is not fixed, (c) and since the glance is not fixed and the bird’s tweet rarely attract my attention, the probability that I believe that there’s a bird given that there’s one is low, and (d) yet I know that there’s a bird when I glance.

Julien, so we have two versions of the case, one where the bird’s activity is probabilistically independent of the glimpse, and the other where it raises the likelihood of the particular glance in question by cause a shift in attention. I described what happens in the first case in the post, so I’ll say something about the second.

In the second kind of case, I’m not sure we need any fancy explanation. It looks more like just an ordinary case of perception: the world gets our attention in some way, and we form a belief on the basis of whatever sensory information is received. So, there’s a causal link from the state of the world to a signal, then to our attention, and finally to our perceptual belief. In the usual case of such causal stories, we won’t get strong enough violations of probabilistic transitivity to have to worry that the adherence probabilities won’t work out right. The only time we need to worry is when it is happenstance that we look, because then there is no apparent link from the state of the world to our belief. So, I think the answer to your second kind of glimpse case is that it turns it into just an ordinary case of perceptual knowledge, and if there’s a problem here, there would be a problem for almost every case of ordinary perceptual belief. Does that seem right to you?

Jon is right about my view, that * fixes the glimpse. (** is the one used for adherence, but that also should have been less than or equals.) The second kind of case just proposed, where we add “silently,” is the reason the conditions are to be applied only to atomic sentences (whatever that means in your chosen language). Adding “silently” is like adding “and the bird was silent.” The conditions aren’t well-defined if you allow applications to any old complexity of sentence so everything gets determined by the atomics. This is important because otherwise the conditions wouldn’t be a solution to the generality problem (which was their purpose). It also has the interesting implication which I noticed recently that you pretty much can’t track conjunctions. The only way you can know them is by tracking conjuncts and conjoining. I don’t think that’s odd at all, and it avoids some longstanding problems for the tracking theory about conjunction.

My explicit response in the book to broadly glimpse-type cases (where the fact that you got the look was an accident) was a response to the Jesse James case using closure. I still think that works (and is more widely applicable than might appear since on my view knowledge of logical implication doesn’t require even ability to make an inference). But I since noticed that the fixing conditions do a lot of work in this area. They also have sensible implications about whether in a given case the method of belief-forming should be fixed or vary. These conditions fix anything probabilistically irrelevant to p, so if you were no more or less likely to use a method because of p’s being true or false, then the method is fixed, otherwise not. So the faithful father who correctly believes his son innocent because of a court trial but who would still believe his son innocent by the different method of faith if the court trial had said the son was guilty, doesn’t know because the evaluation doesn’t fix his actual method; which one he uses depends on the guilt or innocence of his son because it depends on the outcome of the court trial, which if fair depends on that guilt or innocence.

The grandmother case — where if the grandson whom she sees standing before her were not well, then the relatives would have prevented him coming to see her — might seem different because there the method she actually uses to come to believe her grandson is well — vision on him standing there — should be fixed. But there is dependence between p vs. -p and q = grandma gets to use vision and -q. However, though there is probabilistic relation it’s symmetric, so the two sides of the criteria * and ** are large but equal. (Temporality and causation don’t play a role in evaluating these probabilities. The correlational aspect mimics causation somewhat but I say this is the real thing, not causation.)

Sorry if the paragraphs are run together. I put extra lines between but they don’t seem to have shown up in the mock-up they’re showing.

Hi Jon and Sherri,

Thanks for your helpful replies!

About the sophisticated glimpse case (and in reply to Jon): it seems clear that there’s a continuum from cases in which the presence of the bird is probabilistically independent of my glancing in its direction, to cases where it is certain that I will glance conditional on their being a bird. It still seems to me that on the “independent” side of the continuum, you get cases where the fixing condition (*) won’t fix my glancing in the direction of the bird, and so the probability of me believing that there is a bird given that there is one is low (probabilistic adherence is violated); but still I know. But if, as Sherri points out, the answer to glimpse cases was that I satisfy tracking (and thus adherence) with respect to some stronger proposition – perhaps the proposition that I’m seeing a bird? -, then it may be superfluous to rely on how the fixing condition (*) applies to that belief (as opposed to the more basic one that entails that one).